On split involutive regular BiHom-Lie superalgebras

The goal of this paper is to examine the structure of split involutive regular BiHom-Lie superalgebras, which can be viewed as the natural generalization of split involutive regular Hom-Lie algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular BiHom-Lie superalgebra {\mathfrak{L}} is of the form {\mathfrak{L}}=U+{\sum }_{\alpha }{I}_{\alpha } with U a subspace of a maximal abelian subalgebra H and any I α , a well-described ideal of {\mathfrak{L}} , satisfying [I α , I β ] = 0 if [α] ≠ [β]. In the case of {\mathfrak{L}} being of maximal length, the simplicity of {\mathfrak{L}} is also characterized in terms of connections of roots.

Computing abelian subalgebras for linear algebras of upper-triangular matrices from an algorithmic perspective

In this paper, the maximal abelian dimension is algorithmically and computationally studied for the Lie algebra hn, of n×n upper-triangular matrices. More concretely, we define an algorithm to compute abelian subalgebras of hn besides programming its implementation with the symbolic computation package MAPLE. The algorithm returns a maximal abelian subalgebra of hn and, hence, its maximal abelian dimension. The order n of the matrices hn is the unique input needed to obtain these subalgebras. Finally, a computational study of the algorithm is presented and we explain and comment some suggestions and comments related to how it works.

QCD breaks Lorentz invariance and colour

In the previous work [A. P. Balachandran and S. Vaidya, Eur. Phys. J. Plus 128, 118 (2013)], we have argued that the algebra of non-Abelian superselection rules is spontaneously broken to its maximal Abelian subalgebra, that is, the algebra generated by its completing commuting set (the two Casimirs, isospin and a basis of its Cartan subalgebra). In this paper, alternative arguments confirming these results are presented. In addition, Lorentz invariance is shown to be broken in quantum chromodynamics (QCD), just as it is in quantum electrodynamics (QED). The experimental consequences of these results include fuzzy mass and spin shells of coloured particles like quarks, and decay life times which depend on the frame of observation [D. Buchholz, Phys. Lett. B 174, 331 (1986); D. Buchholz and K. Fredenhagen, Commun. Math. Phys. 84, 1 (1982; J. Fröhlich, G. Morchio and F. Strocchi, Phys. Lett. B 89, 61 (1979); A. P. Balachandran, S. Kürkçüoğlu, A. R. de Queiroz and S. Vaidya, Eur. Phys. J. C 75, 89 (2015); A. P. Balachandran, S. Kürkçüoğlu and A. R. de Queiroz, Mod. Phys. Lett. A 28, 1350028 (2013)]. In a paper under preparation, these results are extended to the ADM Poincaré group and the local Lorentz group of frames. The renormalisation of the ADM energy by infrared gravitons is also studied and estimated.

PERIODIC -GRAPH ALGEBRAS REVISITED

Let $P$ be a finitely generated cancellative abelian monoid. A $P$-graph ${\rm\Lambda}$ is a natural generalization of a $k$-graph. A pullback of ${\rm\Lambda}$ is constructed by pulling it back over a given monoid morphism to $P$, while a pushout of ${\rm\Lambda}$ is obtained by modding out its periodicity, which is deduced from a natural equivalence relation on ${\rm\Lambda}$. One of our main results in this paper shows that, for some $k$-graphs ${\rm\Lambda}$, ${\rm\Lambda}$ is isomorphic to the pullback of its pushout via a natural quotient map, and that its graph $\text{C}^{\ast }$-algebra can be embedded into the tensor product of the graph $\text{C}^{\ast }$-algebra of its pushout and $\text{C}^{\ast }(\text{Per}\,{\rm\Lambda})$. As a consequence, in this case, the cycline algebra generated by the standard generators corresponding to equivalent pairs is a maximal abelian subalgebra, and there is a faithful conditional expectation from the graph $\text{C}^{\ast }$-algebra onto it.

On conjugacy of maximal abelian subalgebras and the outer automorphism group of the Cuntz algebra

We investigate the structure of the outer automorphism group of the Cuntz algebra and the closely related problem of conjugacy of maximal abelian subalgebras in . In particular, we exhibit an uncountable family of maximal abelian subalgebras, conjugate to the standard maximal abelian subalgebra via Bogolubov automorphisms, that are not inner conjugate to .

Star-Shapedness and K-Orbits in Complex Semisimple Lie Algebras

Given a complex semisimple Lie algebra is a compact real form of g), let be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra , where t is a maximal abelian subalgebra of . Given x ∈ g, we consider π(Ad(K)x), where K is the analytic subgroup G corresponding to , and show that it is star-shaped. The result extends a result of Tsing. We also consider the generalized numerical range f (Ad(K)x), where f is a linear functional on g. We establish the star-shapedness of f (Ad(K)x) for simple Lie algebras of type B.

AUTOMORPHISMS OF THE UHF ALGEBRA THAT DO NOT EXTEND TO THE CUNTZ ALGEBRA

The automorphisms of the canonical core UHF subalgebra ℱn of the Cuntz algebra 𝒪n do not necessarily extend to automorphisms of 𝒪n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras Mn. In that case, necessary and sufficient conditions for the extension property are presented. Also addressed is the problem of extending to 𝒪n the automorphisms of the diagonal 𝒟n, which is a regular maximal abelian subalgebra with Cantor spectrum. In particular, it is shown that there exist product-type automorphisms of 𝒟n that do not extend to (possibly proper) endomorphisms of 𝒪n.

Hochschild cohomology of II1 factors with Cartan maximal abelian subalgebras

In this paper we prove that, for a type-II1 factor N with a Cartan maximal abelian subalgebra, the Hochschild cohomology groups Hn(N,N)=0 for all n≥1. This generalizes the result of Sinclair and Smith, who proved this for all N having a separable predual.

How to Observe Quantum Fields and Recover Them From Observational Data?

On the basis of the mathematical notion of “micro-macro duality” for understanding mutual relations between microsopic quantum systems (micro) and their macroscopic manifestations (macro) in terms of the notion of sectors and order parameters, a general mathematical scheme is proposed for detecting the state-structure inside of a sector through measurement processes of a maximal abelian subalgebra of the algebra of observables. For this purpose, the Kac-Takesaki operators controlling group duality play essential roles, which naturally leads to the composite system of the observed system and the measuring system formulated by a crossed product. This construction of composite systems will be shown to make it possible for the micro to be reconstructed from its observational data as macro in the light of the Takesaki duality for crossed products.

NONCOMMUTATIVE KdV HIERARCHY

The noncommutative version of the Korteweg–de Vries equation studied here is shown to admit infinitely many constants of motion and to give rise to a hierarchy of higher-order Hamiltonian evolution equations, each one the noncommutative version of the commutative KdV equation of the same order. The noncommutative KdV polynomials span, topologically, a maximal Abelian subalgebra of the Lie algebra of noncommutative Bäcklund transformations. Two classes of examples of "completely integrable" systems of evolution equations to which the theory applies are described in the last two sections.